This project is funded by the DFG within the Priority Programme on the Theoretical Foundations of Deep Learning
Scope. We study product spaces of elementary Riemannian manifolds for the context-sensitive analysis of data observed in any metric space. State spaces interact dynamically by geometric averaging and locally according to the adjacency structure of an underlying graph. The corresponding interaction parameters are learned from data. Geometric integration of the resulting continuous-time flow generates layers of a neural network. Our approach enables to study dynamical relations of inference and learning in neural networks from a geometric viewpoint, along with a probabilistic interpretation of contextual decision making. From the numerical point of view, the approach copes with high dimensions and large problem sizes.
Mathematical aspects. Information geometry, coupled and regularized information transport, geometric mechanics on manifolds and variational principles, geometric numerical integration, statistical performance characterization using PAC-Bayesian analysis.
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